1995 College Scholastic Ability Test

Mathematics·Studies (I)

Hum. & Arts
Let \(\alpha\) and \(\beta\) be the two solutions to the quadratic equation \(2x^2-4x-1=0\). What is the value of \(\alpha^3 + \beta^3\)? [1 point]
  1. \(1\)
  2. \(3\)
  3. \(4\)
  4. \(8\)
  5. \(11\)
Let \(\alpha\) be the solution to the equation \(3^{x+2}=96\). Which option below is correct? [1 point]
  1. \(0 < \alpha < 1\)
  2. \(1 < \alpha < 2\)
  3. \(2 < \alpha < 3\)
  4. \(3 < \alpha < 4\)
  5. \(4 < \alpha < 5\)
\(2\times 2\) matrices \(A\) and \(B\) are as follows.
\(A=\begin{pmatrix*}[r] 2&-4 \\ -1&2 \end{pmatrix*},\; B=\begin{pmatrix} 1&2 \\ 2&4 \end{pmatrix}\)
Which option is equal to the matrix \(\dfrac{1}{3}AB-BA\)? [1 point]
  1. \(\begin{pmatrix*}[r] -2&-4 \\ 1&2 \end{pmatrix*}\)
  2. \(\begin{pmatrix*}[r] -2&8 \\ 2&-4 \end{pmatrix*}\)
  3. \(\begin{pmatrix*}[r] -4&-8 \\ 2&4 \end{pmatrix*}\)
  4. \(\begin{pmatrix*}[r] -6&-12 \\ 3&6 \end{pmatrix*}\)
  5. \(\begin{pmatrix*}[r] 0&0 \\ 0&0 \end{pmatrix*}\)
What is the value of the integral \(\displaystyle\int_{0}^{3}|x-1|dx\)? [1 point]
  1. \(1\)
  2. \(\dfrac{3}{2}\)
  3. \(2\)
  4. \(\dfrac{5}{2}\)
  5. \(3\)

Mathematics·Studies (I)

Hum. & Arts
For the universe \(U\) and its two subsets \(A\) and \(B\), given that \(A \subseteq B\), which option below is not always true? (※ \(U\ne\varnothing\)) [1 point]
  1. \(A\cup B=B\)
  2. \(A\cap B=A\)
  3. \((A\cap B)^c=B^c\)
  4. \(B^c \subseteq A^c\)
  5. \(A-B=\varnothing\)
Let \(f(x)=2x-1\). A function \(g(x)\) satisfies
\((h \circ g \circ f)(x)=h(x)\)
for all functions \(h(x)\). What is the value of \(g(3)\)?
(※ \(f(x), g(x)\) and \(h(x)\) are functions from \(R\) to \(R\), where \(R\) is the set of all real numbers.) [1 point]
  1. \(-2\)
  2. \(-1\)
  3. \(0\)
  4. \(1\)
  5. \(2\)
As the figure shows, there are \(7\) points on a semicircle. What is the number of triangles that have three of these points as its vertices? [1 point]
  1. \(34\)
  2. \(33\)
  3. \(32\)
  4. \(31\)
  5. \(30\)
Suppose
\(\dfrac{1}{(x-1)(x-2)\cdots(x-10)}\)
\(=\dfrac{a_1}{x-1} + \dfrac{a_2}{x-2} + \cdots + \dfrac{a_{10}}{x-10}\)
is satisfied for all real numbers \(x\) that does not make the denominator \(0\). What is the value of \(a_1+a_2+\cdots+a_{10}\)? [1.5 points]
  1. \(0\)
  2. \(-1\)
  3. \(1\)
  4. \(-10\)
  5. \(10\)

Mathematics·Studies (I)

Hum. & Arts
\(x\) and \(y\) are real numbers that satisfy \((x+y)(x-y)\ne 0\) and
\(\sqrt{\dfrac{x+y}{x-y}}=-\dfrac{\sqrt{x+y}}{\sqrt{x-y}}\).
Which option below depicts the region on the \(xy\)-plane where the point \((x, y)\) can exist? (※ The dotted lines are not included.) [1 point]
The instantaneous rate of change of the function \(f(x)\) at \(x=a\) is \(2\). If a differentiable function \(g(x)\) satisfies
\(\displaystyle\lim_{h\to\,0}\!\dfrac{f(a+2h)-f(a)-g(h)}{h}=0\),
what is the value of \(\displaystyle\lim_{h\to\,0}\!\dfrac{g(h)}{h}\)? [1.5 points]
  1. \(4\)
  2. \(2\)
  3. \(0\)
  4. \(-2\)
  5. \(-4\)
A point \(\mathrm{P}\) starts from the origin and moves on the number line for \(7\) seconds, and its velocity \(v(t)\) at time \(t\) is as the figure shows. Which option contains every correct statement in the <List>? [1.5 points]
  1. After point \(\mathrm{P}\) started moving, there was a moment where it stopped for \(1\) second.
  2. While point \(\mathrm{P}\) was moving, it changed its direction \(4\) times.
  3. Point \(\mathrm{P}\) was on the starting point
    \(4\) seconds after it started moving.
  1. a
  2. c
  3. a, b
  4. a, c
  5. b, c
† Rephrasing statement a.: “Between \(\boldsymbol{0\leq t \leq 7}\), there was a moment where point \(\mathrm{P}\) stopped for \(1\) second.”
For all probability density functions \(f(x)\) and \(g(x)\) defined on the closed interval \([0,1]\), which option below can be a probability density function? [1 point]
  1. \(f(x)-g(x)\)
  2. \(f(x)+g(x)\)
  3. \(\dfrac{1}{2}\{f(x)-g(x)\}\)
  4. \(\dfrac{1}{3}\{2f(x)+g(x)\}\)
  5. \(2f(x)-g(x)\)
† No options were correct in the source due to a typo.

Mathematics·Studies (I)

Hum. & Arts
For two square matrices \(A\) and \(B\) of order \(2\) that satisfy
\(A^2+A=E\) and \(AB=2E\),
what is \(B^2\) in terms of \(A\) and \(E\)? (※ \(E\) is the identity matrix of size \(2\).) [1.5 points]
  1. \(2A+4E\)
  2. \(2A-E\)
  3. \(4A+8E\)
  4. \(4A-8E\)
  5. \(8A-4E\)
What is the value of \(m, n\) and \(k\) printed by the following flowchart, respectively? [1.5 points]
  1. \(0,2\) and \(5\)
  2. \(0,2\) and \(6\)
  3. \(0,5\) and \(3\)
  4. \(2,3\) and \(6\)
  5. \(2,3\) and \(5\)
\(18\) identical cube-shaped glass boxes were stacked to make a rectangular cuboid as shown to the right. Some of these boxes were replaced with a black box with the same size. Let \((\alpha)\) be the shape of this cuboid seen from above the rectangle \(\mathrm{ABCD}\), and let \((\beta)\) be the shape of this cuboid seen from the front of the rectangle \(\mathrm{BEFC}\). \((\alpha)\) and \((\beta)\) are as follows.
What is the shape of this cuboid seen from the front of the rectangle \(\mathrm{CFGD}\)? [1.5 points]
Two events \(A\) and \(B\) are subsets of a sample space \(S\) and satisfy \(\mathrm{P}(A)\ne0\) and \(\mathrm{P}(B)\ne0\). Which option contains every correct statement in the <List>? [1.5 points]
  1. If \(A\) and \(B\) are independent, then \(\mathrm{P}(A|B)\) and \(\mathrm{P}(B|A)\) are the same.
  2. If \(A\) and \(B\) are mutually exclusive, then \(\mathrm{P}(A)+ \mathrm{P}(B)\leq 1\).
  3. If \(\mathrm{P}(A\cup B)=1\), then \(B\) is a complement of \(A\).
  1. a
  2. b
  3. a, c
  4. b, c
  5. a, b, c

Mathematics·Studies (I)

Hum. & Arts
The annual income of an employee is \(Y\) (won). Tax is not imposed for \(a\%\) of this income, and a tax of \(b\%\) is imposed for the rest of the income. After paying the taxes, this person spends \(C\) (won) and deposits all of the rest of the after-tax income. What is the annual deposit \(S\) (won) of this person? [1 point]
  1. \(S= \left(\!1- \dfrac{a}{100}- \dfrac{b}{100}\!\right)Y-C\)
  2. \(S= \left(\!1- \dfrac{a}{100}- \dfrac{b}{100}\!\right)Y+C\)
  3. \(S= \left(\!1- \dfrac{a}{100}\cdot \dfrac{b}{100} - \dfrac{b}{100}\!\right)Y-C\)
  4. \(S= \left(\!1+ \dfrac{a}{100}\cdot \dfrac{b}{100} - \dfrac{b}{100}\!\right)Y+C\)
  5. \(S= \left(\!1+ \dfrac{a}{100}\cdot \dfrac{b}{100} - \dfrac{b}{100}\!\right)Y-C\)
Figure below shows the graph of the function \(y=f(x)\). Which option below lists the number of distinct real solutions, and the sum of distinct real solutions, of the equation \(f(f(x+2))=4\) solved for \(x\)? (※ \(f(x)<0\) when \(x<2\) or \(x>19\).) [1.5 points]
  1. \(2,20\)
  2. \(2,22\)
  3. \(3,30\)
  4. \(4,42\)
  5. \(4,50\)
Let us represent a positive integer \(n\) as \(n=2^p \cdot k\)
(\(p\) is a nonnegative integer, \(k\) is an odd number), and let \(f(n)=p\). For example, \(f(12)=2\). Which option contains every correct statement in the <List>? [1 point]
  1. If \(n\) is an odd number, then \(f(n)=0\).
  2. \(f(8)<f(24)\).
  3. There are infinitely many positive integers \(n\) that satisfy \(f(n)=3\).
  1. a
  2. b
  3. a, b
  4. a, c
  5. b, c
Let the set \(U=\{1,2,3,4,\cdots,100\}\). Among sets \(A\) that are a subset of \(U\) and satisfies the following conditions (A) and (B), what is the set with the smallest number of elements? [1 point]
  1. \(3\in A\)
  2. If \(m, n\in A\) and \(m+n \in U\), then \(m+n \in A\).
  1. \(A=\{3,9,15,21,\cdots,99\}\)
  2. \(A=\{3,6,9,12,\cdots,99\}\)
  3. \(A=\{3,4,5,6,\cdots,100\}\)
  4. \(A=\{1,3,5,7,\cdots,99\}\)
  5. \(A=\{1,2,3,4,\cdots,100\}\)

Mathematics·Studies (I)

Hum. & Arts
As the figure shows, there is a trapezoid \(\mathrm{ABCD}\).
\(\overline{\mathrm{AB}}=\overline{\mathrm{AD}}=1\), \(\:\overline{\mathrm{BC}}=2\),
and the magnitude of \(\angle \mathrm{A}\) and \(\angle \mathrm{B}\) are \(\dfrac{\pi}{2}\).
Let \(\mathrm{P}\) be a point on the top edge \(\mathrm{AD}\) and let \(\overline{\mathrm{PB}}=x\) and \(\overline{\mathrm{PC}}=y\). Which option contains every correct statement in the <List>? [1.5 points]
  1. \(xy\geq 2\).
  2. If \(xy=2\), then \(\triangle \mathrm{BCP}\) is a right triangle.
  3. \(xy\leq \sqrt{5}\).
  1. a
  2. c
  3. a, c
  4. b, c
  5. a, b, c
The following is a proof of the law of cosines relating the side lengths of a triangle to the cosine of angles, specifically if \(\angle \mathrm{A}\) is obtuse in \(\triangle \mathrm{ABC}\).
(Proof) As the figure shows, let \(\triangle \mathrm{ABC}\) with side lengths \(a, b\) and \(c\) be on the \(xy\)-plane so that point \(\mathrm{A}\) is on the origin. Let \((x,y)\) be the coordinates of point \(\mathrm{C}\). Then
\(x= \fbox{\(\;(\alpha)\;\)}\) and \(y= \fbox{\(\;(\beta)\;\)}\),
so the following holds by the Pythagorean theorem.
\(\begin{align}a^2 &= (\fbox{\(\;(\gamma)\;\)})^2+y^2\\ &= b^2+c^2-2bc \cos A \end{align}\)
In the proof above, what are appropriate for \((\alpha), (\beta)\) and \((\gamma)\) in this order? [1 point]
① \(b\cos A\), \(b\sin A\), \(c+x\)
② \(b\cos A\), \(b\sin A\), \(c-x\)
③ \(b\cos A\), \(-b\sin A\), \(c+x\)
④ \(-b\cos A\), \(-b\sin A\), \(c-x\)
⑤ \(-b\cos A\), \(-b\sin A\), \(c+x\)
For a cubic equation \(x^3+ax^2+bx+c=0\) with three real solutions, let \(\alpha, \beta\) and \(\gamma\) be the three solutions. The following is a proof that at least one of the three solutions has an absolute value greater than or equal to \(\dfrac{|a|}{3}\).
(Proof)
Let us assume the conclusion is false and \(\fbox{ (A) }\). Then
\(|\,\alpha\,|< \dfrac{|a|}{3},\; |\,\beta\,|< \dfrac{|a|}{3}\:\) and \(\:|\,\gamma\,|< \dfrac{|a|}{3}\).
Using the relation between solutions and coefficients,
\(a= \fbox{ (B) }\),
and
\(\begin{align} |a| &\leq |\alpha+\beta| + |\gamma| \\ &\leq \fbox{ (C) }\\ &< \dfrac{|a|}{3}+\dfrac{|a|}{3}+\dfrac{|a|}{3}=|a|. \end{align}\)
However, this contradicts the premise. Therefore at least one of the three solutions has an absolute value greater than or equal to \(\dfrac{|a|}{3}\).
In the proof above, what are appropriate for \(\text{(A)}, \text{(B)}\) and \(\text{(C)}\) in this order? [1 point]
  1. some solutions have an absolute value less than \(\dfrac{|a|}{3}\),
    \(-(\alpha+\beta+\gamma),\quad |\alpha|+|\beta|+|\gamma|\)
  2. some solutions have an absolute value less than or equal to \(\dfrac{|a|}{3}\),
    \(\alpha+\beta+\gamma,\quad |\alpha|+|\beta|+|\gamma|\)
  3. all solutions have an absolute value less than \(\dfrac{|a|}{3}\),
    \(\alpha+\beta+\gamma,\quad |\alpha+\beta+\gamma|\)
  4. some solutions have an absolute value less than \(\dfrac{|a|}{3}\),
    \(-(\alpha+\beta+\gamma),\quad |\alpha|+|\beta|+|\gamma|\)
  5. all solutions have an absolute value less than or equal to \(\dfrac{|a|}{3}\),
    \(\alpha+\beta+\gamma,\quad |\alpha+\beta+\gamma|\)

Mathematics·Studies (I)

Hum. & Arts
The following is a proof of a theorem about the harmonic mean.
(Proof) For positive numbers \(a,b\) and \(H\), suppose a real number \(r\) exists such that
\(a=H+\dfrac{a}{r}\) and \(H=b+\dfrac{b}{r}\)
\((A)\)
is satisfied. Then \(a\ne b\) and
\(\dfrac{a-H}{a}= \fbox{\(\;(\alpha)\;\)}\)
\((B)\)
therefore \(H=\fbox{\(\;(\beta)\;\)}\).
Conversely, for positive numbers \(a\) and \(b\) where \(a\ne b\), if \(H=\fbox{\(\;(\beta)\;\)}\), then the expression \((B)\) holds and \(\dfrac{a-H}{a}\ne 0\).
From \((B)\), let \(\dfrac{a-H}{a}=\dfrac{1}{r}\). Then the expression \((A)\) holds.
Therefore, for positive numbers \(a, b\) and \(H\),
‘\(a\ne b\) and \(H=\fbox{\(\;(\beta)\;\)}\)’ is \(\fbox{\(\;(\gamma)\;\)}\) for a real number \(r\) to exist such that expression \((A)\) holds.
In the proof above, what are appropriate for \((\alpha), (\beta)\) and \((\gamma)\) in this order? [1.5 points]
  1. \(\dfrac{H-b}{b}\), \(\dfrac{2ab}{a+b}\), necessary and sufficient
  2. \(\dfrac{H-b}{b}\), \(\dfrac{ab}{a+b}\), necessary and sufficient
  3. \(\dfrac{H-b}{b}\), \(\dfrac{2ab}{a+b}\), sufficient
  4. \(\dfrac{b-H}{b}\), \(\dfrac{2ab}{a+b}\), necessary
  5. \(\dfrac{b-H}{b}\), \(\dfrac{ab}{a+b}\), sufficient
For all positive numbers \(n\), the polynomial function \(f_n(x)\) has the following properties.
  1. \(f_1(x)=x^2\)
  2. \(f_{n+1}(x)=f_n(x)+f_n'(x)\)
What is the constant term of \(f_{25}(x)\)? [1.5 points]
  1. \(548\)
  2. \(550\)
  3. \(552\)
  4. \(554\)
  5. \(556\)
On the \(xy\)-plane, there are two points \(\mathrm{O}(0,0)\) and \(\mathrm{A}(2,0)\), and a point \(\mathrm{P}(t,2)\) moving on the line \(y=2\). Let \(\mathrm{Q}\) be the point where line \(\mathrm{AP}\) meets the line \(y=\dfrac{1}{2}x\). Let \(t_1\) be the value of \(t\) for which the area of \(\triangle\mathrm{QOA}\) divided by the area of \(\triangle\mathrm{POA}\) is equal to \(\dfrac{1}{3}\),
let \(t_2\) be the value of \(t\) for which it is equal to \(\dfrac{1}{2}\), \(\cdots\), let \(t_n\) be the value of \(t\) for which it is equal to \(\dfrac{n}{n+2}\). What is the value of \(\displaystyle\lim_{n\to\,\infty}t_n\)? [2 points]
  1. \(0\)
  2. \(1\)
  3. \(2\)
  4. \(3\)
  5. \(4\)
What is the maximum value of the function \(f(x)=\log_9 (5-x) + \log_3 (x+4)\)? [1.5 points]
  1. \(\dfrac{7}{2}\)
  2. \(4\)
  3. \(\dfrac{2}{5}+\log_3 4\)
  4. \(\dfrac{3}{2}+\log_3 2\)
  5. \(4+\log_3 6\)

Mathematics·Studies (I)

Hum. & Arts
As the figure shows, Let \(\mathrm{A_1, A_2,}\cdots\) be points on the ray \(\mathrm{OA}\), and let \(\mathrm{B_1, B_2,}\cdots\) be points on the ray \(\mathrm{OB}\), such that
\(\overline{\mathrm{OA_1}}=\overline{\mathrm{A_1B_1}}=\overline{\mathrm{B_1A_2}}=\cdots\).
It is given that using this method, we can create four isosceles triangles
\(\triangle\mathrm{OA_1B_1}, \triangle\mathrm{A_1B_1A_2}, \triangle\mathrm{B_1A_2B_2}, \triangle\mathrm{A_2B_2A_3}\)
but we cannot create the fifth isosceles triangle. Let \(\theta\) be the magnitude of \(\angle \mathrm{AOB}\). What is the range of \(\theta\)? [2 points]
  1. \(\dfrac{\pi}{4} \leq\theta\leq \dfrac{\pi}{2}\)
  2. \(\dfrac{\pi}{7} \leq\theta\leq \dfrac{\pi}{5}\)
  3. \(\dfrac{\pi}{10} \leq\theta\leq \dfrac{\pi}{8}\)
  4. \(\dfrac{\pi}{14} \leq\theta\leq \dfrac{\pi}{12}\)
  5. \(\dfrac{\pi}{17} \leq\theta\leq \dfrac{\pi}{15}\)
Let \(x\) be the amount of labor input and \(y\) be the amount of capital input for some industry. Then, the output \(z\) of the industry is known to be as follows.
\(z=2x^\alpha y^{1-\alpha}\) (\(\alpha\) is a constant with \(0<\alpha<1\))
According to some data, the amount of labor input and capital input in the year \(1993\) was \(4\) times and \(2\) times that of the year \(1980\), respectively, and the industry output in the year \(1993\) was \(2.5\) times that of the year \(1980\). From this data, what is the value of the constant \(\alpha\) calculated to the hundredth?
(※ \(\log_{10}2=0.30\)) [2 points]
  1. \(0.50\)
  2. \(0.33\)
  3. \(0.25\)
  4. \(0.20\)
  5. \(0.10\)
A fully automated shelf factory produces two kinds of shelves, ‘\(X\)’ and ‘\(Y\)’. Two machines \(\mathrm{A}\) and \(\mathrm{B}\) are used to produce these shelves. For safety reasons, machines \(\mathrm{A}\) and \(\mathrm{B}\) cannot be operated for more than \(18\) hours a day and \(20\) hours a day, respectively. To produce \(1\) unit of shelf ‘\(X\),’ machines \(\mathrm{A}\) and \(\mathrm{B}\) must be operated for \(3\) hours and \(5\) hours respectively, and to produce \(1\) unit of shelf ‘\(Y\),’ machines \(\mathrm{A}\) and \(\mathrm{B}\) must be operated for \(6\) hours and \(5\) hours respectively. A unit of shelf ‘\(X\)’ and shelf ‘\(Y\)’ are sold for \(2\) million won and \(3\) million won respectively. Assuming that shelves that are produced are instantly sold, what is the maximum sales for one day? [2 points]
Shelf ‘\(X\)’ Shelf ‘\(Y\)’ Operating
time limit
Machine \(\mathrm{A}\) \(3\) hours \(6\) hours \(18\) hours
Machine \(\mathrm{B}\) \(5\) hours \(5\) hours \(20\) hours
Selling
price		 \(2\) million
won		 \(3\) million
won
  1. \(9\) million won
  2. \(10\) million won
  3. \(11\) million won
  4. \(12\) million won
  5. \(13\) million won